Quantum tunnelling.

Step 1 of 5

A wall too high

A ball rolling at a hill too high for it always comes back. In classical physics there is no other way: halfway up, its kinetic energy would have to be negative. But we saw that an electron has a wave, and the wave decides where it can be found.

On the bench an electron with an energy of 1 eV runs at a barrier: up is energy, along is distance. The electron is a wave packet that travels, spreads and splits, and the tags show how much of it bounced back and how much got through. Right now Wall height is 0.5 eV, below its energy, and Wall thickness is fixed. Predict what will happen when the wall is higher than the electron.

Step 2 of 5

Inside the wall

Inside a wall higher than the electron’s energy the wave does not wave: it fades, an exponential decay. The wave itself fades as $e^{-\kappa x}$⁠, and the chance of finding the electron, the square of the wave, as $e^{-2\kappa x}$⁠, where

$\kappa = \frac{\sqrt{2m(V - E)}}{\hbar}$$\kappa =$$\frac{\sqrt{2m(V - E)}}{\hbar}$⁠

The higher the wall above the electron, the larger $\kappa$⁠ and the faster the wave fades. Wall height, Wall thickness and Look into the wall are fixed.

Try it: after checking, you can step Look into the wall to Inside the wall in the next Step, and see the fading wave and the depth at which the chance halves.

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Step 3 of 5

Thicker walls

A thick wall, and inside it the wave fades further and further before it reaches the far side. What gets through a thick wall shrinks roughly as $e^{-2\kappa d}$⁠, where $d$⁠ is the thickness, so every fixed extra thickness divides it by a fixed factor. Wall height is fixed. Change Wall thickness.

Try it: after checking, step Look into the wall to Thicker walls. The chart beside the Scene shows what gets through against the thickness on a log scale: a straight line.

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Step 4 of 5

Higher walls

The height enters the exponent too, through $\kappa$⁠. Wall thickness is fixed. Predict what will happen when the gap between the wall and the electron grows threefold.

Step 5 of 5

The far side

At Inside the wall the Scene shows the steady wave at the electron’s energy: before the wall it is large, inside it fades, and after it carries on, smaller.

Quantum tunnelling is no laboratory curiosity. It is how an alpha particle escapes a nucleus in alpha decay, how hydrogen nuclei in the Sun get close enough to fuse, and how a scanning tunnelling microscope feels a surface: the current between the tip and the surface changes tenfold when the gap changes by less than an atom’s width.